Wallace wore the armband at Rangers for a number of years, but with his future at the Glasgow club now uncertain, Steven Gerrard has passed the captaincy on.
James Tavernier now wears the armband for Rangers, which Miller suggests is a deserved honour for the full-back.
However, Miller was left disappointed by Gerrard’s decision to take the armband off his former Rangers teammate, Wallace.
He said: “I’m disappointed for Lee. He’s carried that role with real pride and I know how much it meant for him to wear that armband every week. It’s disappointing in that respect, but I’m delighted for James because he deserves it.”
Wallace fell out of favour at Rangers last season when Graeme Murty was in charge, and Gerrard is yet to give him a fresh chance.
Wallace was left off Rangers’ pre-season tour, and did not feature in the squad for the Light Blues’ match against Shkupi last night.
Tavernier scored during his first match wearing the Rangers armband, as Gerrard’s side won 2-0 in the Europa League last night.